# Amplitudes, recursion relations and unitarity in the Abelian Higgs Model

## TL;DR

This paper proves tree-level unitarity in the Abelian Higgs model using diagrammatic and recursion relation methods, demonstrating cancellations of high-energy divergences and establishing partial-wave unitarity for all on-shell amplitudes.

## Contribution

It introduces combinatorial and on-shell recursion relations for amplitudes in the Abelian Higgs model, providing a novel proof of unitarity at tree level.

## Key findings

- Tree-level unitarity is proven inside the unitary gauge.
- High-energy divergences cancel at the first two orders for any tree amplitude.
- On-shell recursion relations confirm partial-wave unitarity for all amplitudes.

## Abstract

The Abelian Higgs model forms an essential part of the electroweak standard model: it is the sector containing only Z and Higgs bosons. We present a diagram-based proof of the tree-level unitarity of this model inside the unitary gauge, where only physical degrees of freedom occur. We derive combinatorial recursion relations for off-shell amplitudes in the massless approximation, which allows us to prove the cancellation of the first two orders in energy of unitarity-violating high-energy behaviour for any tree-level amplitude in this model. We describe a deformation of the amplitudes by extending the physical phase space to at least 7 spacetime dimensions, which leads to on-shell recursion relations a la BCFW. These lead to a simple proof that all on-shell tree amplitudes obey partial-wave unitarity.